| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2026 |
| Session | November |
| Marks | 9 |
| Topic | Polar coordinates |
| Type | Show polar curve has Cartesian form |
| Difficulty | Challenging +1.2 This is a standard Further Maths polar coordinates question requiring conversion from Cartesian form, finding a maximum using calculus, and computing an area integral. While it involves multiple steps and FM content, the techniques are routine: substituting x=r cos θ, y=r sin θ, differentiating to find maxima, and applying the standard polar area formula. The algebraic manipulation is straightforward and the question structure is typical of textbook exercises. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
The figure below shows the curve with cartesian equation $(x^2 + y^2)^2 = xy$.
\includegraphics{figure_3}
\begin{enumerate}[label=(\alph*)]
\item Show that the polar equation of the curve is $r^2 = a \sin b\theta$, where $a$ and $b$ are positive constants to be determined. [3]
\item Determine the exact maximum value of $r$. [2]
\item Determine the area enclosed by one of the loops. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2026 Q3 [9]}}